Power-law singularity in the local density of states due to the point defect in graphene
Wen-Min Huang,
1
Jian-Ming Tang,
2
and Hsiu-Hau Lin
1,3
1
Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan
2
Department of Physics, University of New Hampshire, Durham, New Hampshire 03824-3520, USA
3
Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan
Received 25 July 2009; published 17 September 2009
Defects in graphene give rise to zero modes that are often related to the sharp peak in the local density of
states near the defect site. Here we solved all zero modes induced by a single defect in the finite-size graphene
and show that their contributions to the local density of states vanish in the thermodynamic limit. Instead, lots
of resonant states emerge at low energies and eventually lead to a power-law singularity in the local density of
states. Our findings show that the impurity problem in graphene should be treated as a collective phenomenon
rather than a single impurity state.
DOI: 10.1103/PhysRevB.80.121404 PACS numbers: 81.05.Uw, 71.20.-b, 71.23.-k, 71.55.-i
Graphene, a single-layer graphite composed of carbon at-
oms arranged in two-dimensional honeycomb lattice, is re-
cently fabricated in laboratory and attracts intense attentions
from both experimental and theoretical aspects.
1–4
One of the
striking features of graphene is its relativistic description,
described by a pair of massless Dirac fermions in the low-
energy regime.
5,6
The relativistic spectrum gives rise to in-
teresting phenomena such as half-integer quantum Hall
effect,
7
Klein paradox,
8
edge magnetism,
9,10
and others.
11–13
In a two-dimensional Dirac system a peculiar quasilocal-
ized state can be induced by a point defect.
14
Such a state can
be seen in graphene as a pronounced peak in the local den-
sity of states LDOS at zero energy.
15–17
Density-functional
studies
18,19
suggest that point defects can be created by
chemisorption of hydrogen atoms.
20
Furthermore, there
exists quantized magnetic moment associated with each
defect.
18,21,22
Is the peak in the LDOS near the defect site caused by the
zero modes in graphene? It is tempting to say yes. However,
we revisit this problem and find the origin of the peak is not
from the zero modes. We start from the finite-size graphene
in nanotorus geometry and investigate how the system
evolves toward the thermodynamic limit. At finite system
size N, we can solve the zero modes due to a single defect
analytically, contributing a zero-energy peak in LDOS as
expected.
23
However, its spectral weight decays as 1 / ln N or
1 / N depending on whether the nanotorus is semiconducting
or metallic and eventually vanishes in two dimensions.
Therefore, even though the defect gives rise to zero modes,
they do not contribute to the pronounced peak in LDOS for
two-dimensional graphene.
The pronounced peak in LDOS found in previous studies
can be explained in two steps. First of all, our numerics show
that the defect in graphene induces enormous resonant peaks
in the LDOS at energies close to zero. Then, as the system
size grows to infinity, these peaks crowd into zero energy
and become singular. Both numerical and analytic ap-
proaches give 1 / |E| power-law singularity with weak loga-
rithmic corrections. That is to say, the peak in the LDOS is
not from a single impurity state. Instead, it is a power-law
singularity from collective resonance induced by the defect.
It is rather amusing that the impurity state in graphene dis-
solve into a power-law singularity as the single-particle state
disappears in one-dimensional interacting electron gas.
24
The
emergence of the power-law singularity resembles the quan-
tum criticality found in many two-dimensional systems
25,26
and suggest that graphene is a quantum critical system
27,28
as
well. As a result, introducing a point defect reshuffles the
LDOS leading to a power-law singularity rather than a delta-
function or broadened Lorentzian peak.
Now we walk through the details which lead to the con-
clusions sketched in above. Because the band structure of
graphene obtained by the first-principles calculations is well
approximated by the nearest-neighbor hopping for the active
orbitals,
29,30
it is sufficient to start from a tight-binding
Hamiltonian and add a single defect at the origin,
H =- t
r,r'
c
†
rcr' + c
†
r'cr + V
0
c
†
0c0 , 1
where t is the nearest-neighbor hopping amplitude, and V
0
is
the strength of the impurity potential. In the remaining part
of the calculations, we mainly focus on the unitary limit V
0
→ . Though the impurity state in the unitary limit has been
solved analytically in a recent paper,
15
it is insightful to red-
erive it with cares so that the evolution of the impurity state
with the system size N is clarified. For simplicity, we apply
periodic boundary conditions in both directions, wrapping
the finite-size graphene into a nanotorus. The number of unit
cells along the x and y axes is N
x
and N
y
and the lattice sites
are labeled by the coordinates x and y = n + x / 2 with x
=0, 1,..., N
x
/ 2 and n =0,1,..., N
y
-1 as shown in
Fig. 1.
Let us consider the carbon nanotube limit first by taking
N
x
→ but keeping N
y
finite. Because the honeycomb lattice
is bipartite, the wave function of the zero modes only show
up in one of the sublattices
31
as shown in Fig. 1. The wave
function for the left and right sectors can be written as the
linear combinations of all evanescent modes,
l
x, y =
l
C
l
e
ik
l
y
z
l
-x
,
r
x, y =
r
C
r
e
ik
r
y
z
r
x
, 2
PHYSICAL REVIEW B 80, 121404R2009
RAPID COMMUNICATIONS
1098-0121/2009/8012/1214044 ©2009 The American Physical Society 121404-1